First, we need to convert the given angles into their respective trigonometric functions.

arcsinπ/2 = π/2
arccos(-1) = π
arctg√3 = π/3
arcctg(-√3) = -π/6

Now, we add these angles together:

π/2 + π + π/3 - π/6

Combine the angles:

π/2 + 6π/6 + 2π/6 - π/6
= (6π + 2π - π)/6
= 7π/6

So, the result of the expression arcsin(π) + arccos(-1) + arctg(√3) + arcctg(-√3) is 7π/6.